Discrete Subgroups of Lie Groups and Applications to Moduli: Papers Presented at the Bombay Colloquium, 1973, by Baily ... (et al.)

379 Pages·1976·2.592 MB·English

by  Walter L. Baily Jr.

Discrete Subgroups of Lie Groups and
Applications to Moduli

DISCRETE SUBGROUPS OF LIE GROUPS
AND
APPLICATIONS TO MODULI
Paperspresented attheBombayColloquium 1973,by
BAILYFREITAGGARLANDGRIFFITHS
HARDERIHARAMOSTOWMUMFORD
RAGHUNATHANSCHMIDVINBERG
Publishedforthe
TATA INSTITUTE OF FUNDAMENTALRESEARCH,
BOMBAY
OXFORDUNIVERSITY PRESS
1975
OxfordUniversity Press
OXFORDLONDONGLASGOWNEWYORK
TORONTOMELBOURNEWELLINGTONCAPETOWN
DELHIBOMBAYCALCUTTAMADRASKARACHILAHORE
DACCA
KUALALUMPURSINGAPOREJAKARTAHONGKONGTOKYO
NAIROBIDARESSALAAMLUSAKAADDISABABA
IBADANZARIAACCRABEIRUT
(cid:13)c TataInstitute ofFundamental Research,1975
PrintedbyV.B.GharpureatTataPressLimited,414VeerSavarkar
Marg,Bombay400025,andPublishbyC.H.Lewis,Oxford
UniversityPress,ApolloBunder, Bombay400001
PRINTEDININDIA
International Colloquium on Discrete
Subgroups of Lie Groups and Applications
to Moduli
Bombay, 8-15 January 1973
REPORT
ANINTERNATIONALCOLLOQUIUMon‘DiscreteSubgroupsof
Lie Groups and Applications to Moduli’ was held at the Tata Institute
of Fundamental Research, Bombay, from 8 to 15 January 1973. The
purpose oftheColloquium wastodiscuss recentdevelopments insome
aspectsofthefollowingtopics: (i)LatticesinLiegroups,(ii)Arithmetic
groups,automorphicformsandrelatednumber-theoretic questions, (iii)
Moduli problems and discrete groups. The Colloquium was a closed
meetingofexpertsandofothersspecially interested inthesubject.
Th Colloquium was jointly sponsored by the International Mathe-
maticalUnionandtheTataInstitute ofFundamental Research, andwas
financially supported bythemandtheSirDorabjiTataTrust.
AnOrganizing Committeeconsisting ofProfessors A.Borel. M.S.
Narasimhan, M. S. Raghunathan, K. G. Ramananthan and E. Vesentini
was in charge of the scientific programme. Professors A. Borel and
E. Vesentini acted as representatives of the International Mathematical
UnionontheOrganizingCommittee.
The following mathematicians gave invited addresses at the Col-
loquium: W. L. Baily, Jr., E. Freitag, H. Garland, P. A. Griffiths, G.
Harder, Y. Ihara, G. D. Mostow, D. Mumford, M. S. Raghunathan and
W.Schmid.
REPORT
Professor E`.B.Vinberg, whowasunable toattend theColloquium,
sentinapaper.
The invited lectures were of fifty minutes’ duration. These were
followed by discussions. In addition to the programme of invited ad-
dresses, there we expository and survey lectures and lectures by some
invited speakers givingmoredetailsoftheirwork.
The social programme during the Colloquium included a TeaParty
on 8 January; a Violin recital (Classical Indian Music) on 9 January; a
programmeofWesternMusicon10January;aperformanceofClassical
Indian Dances (Bharata Natyan) on 12 January; a Film Show (Pather
Panchali)on13January; andadinnerattheInstitute on14January.
Contents
1. WalterL.Baily,Jr.: Fouriercoefficients of 1–8
Eisenstein seriesontheAdelegroup
2. EberhardFreitag: Automorphyfactors of 9–20
Hilbert’smodulargroup
3. HowardGarland: Onthecohomology ofdiscrete 21–31
subgroups ofsemi-simpleLiegroups
4. PhillipGriffithsandWilfriedSchmid: Recent 32–134
developments inHodgetheory: adiscussion of
techniques andresults
5. G.Harder: Onthecohomology ofdiscrete 135–170
arithmetically definedgroups
6. YasutakaIhara: Onmodularcurvesoverfinite 171–215
fields
7. G.D.Mostow: Strongrigidity ofdiscrete 216–223
subgroups andquasi-conformal mappings overa
division algebra
8. DavidMumford: Anewapproachtocompactifying 224–240
locally symmetricvarieties
9. M.S.Raghunathan: Discretegroupsand 241–343
Q-structures onsemi-simpleLiegroups
10. E.B.Vinberg: Somearithmetical discrete groups 344-372
inLobacˆevskiˆispaces
FOURIER COEFFICIENTS OF
EISENSTEIN SERIES ON THE ADELE
GROUP
By WALTERL.BAILY, JR.
Much of what I wish to present in this lecture will shortly appear 1
elsewhere [3], so for the published part ofthis presentation Ishall con-
finemyselftoarestatementofcertaindefinitionsandresults,concluding
with a few remarks on an area that seems to hold some interest. As in
[3],Iwishtoaddherealsothatmanyoftheactualproofsaretobefound
inthethesisofL.C.Tsao[8].
Let G be a connected, semi-simple, linear algebraic group defined
over Q, which, for simplicity, we assume to be Q-simple (by which
we mean G has no proper, connected, normal subgroups defined over
Q). We assume G to be simply-connected, which implies in particular
that G is connected [2, Ch. 7, §5]. Assume that G has no compact
R R
(connected)simplefactorsandthatifK isamaximalcompactsubgroup
of it, then X = K/G has a G -invariant complex structure, i.e., X is
R R
Hermitian symmetric. Then [6] strong approximation holds forG. We
assume,finally,thatrk (G)(thecommondimensionofallmaximal,Q-
Q
split tori of G) is > 0 and that the Q-relative root system of G is
Q
P
of type C (in the Cartan-Killing classification). Then there exists a to-
tally real algebraic number field k and a connected, almost absolutely
simple, simply-connected algebraic group G′ defined over k such that
G = R G′; therefore, if G is written as a direct product ΠG of al-
k/Q i
mostabsolutely simplefactorsG,theneachG isdefinedoveratotally
i i
real algebraic number field, each G is simply-connected, each G is
i iR
connected and the relative root systems = (G) are of type C
RPi RP i
1
2 WALTERL.BAILY,JR.
[4].
Letting K denote a maximal compact subgroup ofG , the Hermi-
i iR
tiansymmetricspace X = K/G isisomorphic toatubedomainsince
i i iR
is of typeC [7], hence X = Π X is bi-holomorphically equivalent
RPi i i
toatubedomain
T= {Z = X+iY ∈ Cn|Y ∈ R},
2 where R is a certain type of open, convex cone in Rn. Let H be the
group of linear affine transformations of T of the form Z 7−→ AZ + B,
where B ∈ Rn, and A is a linear transformation of Rn carrying R onto
itself, and let H˜ be its complete pre-image in G with respect to the
R
natural homomorphism ofG into Hol(T), the group ofbiholomorphic
R
automorphisms of T. Then H˜ = P , where P is an R-parabolic sub-
R
group of G, and from our assumption that is of type C, it follows
QP
that we may assume P to be defined over Q (the reasons for which are
somewhattechnical, butmayallbefoundin[4]).
AssumeG ⊂GL(V),whereV isafinite-dimensional, complexvec-
tor space with a Q-structure. Let Λ be a lattice in V , i.e., a discrete
R
subgroup such that V /Λ is compact, and suppose that Λ ⊂ V . Let
R Q
Γ = {γ ∈ G | γ · Λ = Λ}, and for each finite prime p, let Λ =
Q p
Λ(cid:13) Z ,K = {γ ∈ G | γ·Λ = Λ }. It may be seen, since strong
Z p p Qp p p
approximation holds for G, that K is the closure Γ of Γ in G (in
p p Qp
the ordinary p-adictopology). Nowthe adele groupG ofG isdefined
A
as Π′G , where Π′ denotes restricted direct product with respect to
Qp
the family {K } of compact sub-groups. Define K = K, K∗ = Π K
p ∞ p
p6∞
(Cartesian product).
Forallbutafinitenumberoffinitep,wehaveG = K ·P ,and
Qp p Qp
by changing the lattice Λ at a finite number of places, we may assume
[5] thatG = K ·P for all finite p. In addition, from the Iwasawa
Qp p Qp
decomposition we have G = K · P0, where P0 denotes the identity
R ∞ R R
component of P .
R
We may write the Lie algebra g of G as the direct sum of k , the
C C
complexification oftheLiealgebrakof K,andoftwoAbeliansubalge-
bras p+ and p−, both normalized by k, such that p+ may be indentified
FOURIERCOEFFICIENTSOFEISENSTEINSERIESONTHEADELE
GROUP 3
with Cn ⊃ T. Let K be the analytic subgroup ofG with Lie algebra
C C
k and let P± = exp(p±); then K · P+ is a parabolic subgroup of G
C C
which we may take to be the same as P, and P+ = U is its unipotent
radical. Now p+ has the structure of a Jordan algebra over C, supplied 3
withahomogeneous normformN suchthatAdK iscontained inthe
C
similarity group
S = {g ∈GL(n,C)=GL(p+)|N (gX) = v(g)N (X)}
of N ,where v : S → C× isarational character [7], defined overQif
we arrange things such that K = L is a Q-Levi subgroup of P. (Note
C
that K and L are, respectively, compact and non-compact real forms
R
of KC.) Define v∞ as the character on KC given by v∞(k) = v(Adp+k).
Define v as the character on K given by v (k) = v(Ad +k). If p 6
∞ C ∞ p
∞ let | | be the “standard” p-adic norm, so that the product formula
p
holds. We define (for p 6 ∞) χ on P by χ (ku) = |v(Ad +k)| ,
p Qp p p p
k ∈ L , u ∈ U and χ on P by χ ((p )) = Π χ (p ), which is
Qp Qr A A A p p p p
well defined since for (p ) ∈ P , we have χ (p ) = 1 for all but a
p A p p
finite number of p. Now v is bounded on K and v takes positive real
∞
values on P0, hence v (K ∩ P0) = {1}. Moreover, K is compact and
R ∞ R p
thereforeχ (K ∩P )= {1}. Nowletmbeanypositiveinteger. Define
p p QP
P0 = {(p ) ∈ P |p = 1}, so that P = P , and put P∗ = P0P0.
A p A ∞ A R·P0 A R A
From our previous discussion it is clear that GA = K∗ · P∗. Define the
A A
function ϕ ofG by
m A
ϕ (k∗· p )= v (k )−mχ (p )−m,
m ∗ ∞ ∞ A ∗
where p ∈ P∗, k∗ ∈ K∗, k∗ = (k ). It follows from the preceding that
∗ A p
ϕ iswelldefined.
m
Bytheproduct formula,χ (p) = 1for p= p . Define
A Q
E˜ (g) = ϕ (gγ), g ∈G .
m X m A
γ∈GQ/PQ
Byacriterion of Godement, this converges normally onG ifmissuf-
A
ficientlylarge.